1.206J/16.77J/ESD.215J Airline Schedule Planning Cynthia Barnhart Spring 2003

1.206J/16.77J/ESD.215J Airline Schedule Planning: Multi-commodity Flows Outline Applications Problem definition Formulations Solutions Computational results Integer multi-commodity network flow problems Integer multi-commodity network flow solutions Branch-and-price: combination of branch-and-bound and column generation Results 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Application I Package flow problem (express package delivery operation) Shipments have specific origins and destinations and must be routed over a transportation network Each set of packages with a common origin-destination pair is called a commodity Time windows (availability and delivery time) associated with packages The objective might be to minimize total costs, find a feasible flow, ... 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Application II Passenger mix problem Given a fixed schedule of flights, a fixed fleet assignment and a set of customer demands for air travel service on this fleeted schedule, the airline's objective is to maximize revenues by accommodating as many high fare passengers as possible For some flights, demand exceeds seat supply and passengers must be spilled to other itineraries of either the same or another airline 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Application III Message routing problem In a telecommunications or computer network, requirements exist for transmission lines and message requests, or commodities. The problem is to route the messages from their origins to their respective destinations at minimum cost 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

MCF Networks Set of nodes Set of arcs Each node associated with the supply of or demand for commodities Set of arcs Cost per unit commodity flow Capacity limiting the total flow of all commodities and/ or the flow of individual commodities 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

MCF Commodity Definitions A commodity may originate at a subset of nodes in the network and be destined for another subset of nodes A commodity may originate at a single node and be destined for a subset of the nodes A commodity may originate at a single node and be destined for a single node 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

MCF Objectives Flow the commodities through the networks from their respective origins to their respective destinations at minimum cost Expressed as distance, money, time, etc. Ahuja, Magnanti and Orlin (1993)-- survey of multi-commodity flow models and solution procedures 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

MCF Problem Formulations -- Linear Programs Network flow problems Capacity constraints limit flow of individual commodities Conservation of flow constraints ensure flow balance for individual commodities Flow non-negativity constraints With side constraints Bundle constraints restrict total flow of ALL commodities on an arc 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

MCF Constraint Matrix Network flow problem, commodity k=1 Bundle constraints limiting total flow of all commodities to arc capacities 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Alternative Formulations for O-D Commodity Case Node-Arc Formulation Decision variables: flow of commodity k on each arc ij Path Formulation Decision variables: flow of commodity k on each path for k “Tree” or “Sub-network” Formulation Define: super commodity: set of all (O-D) commodities with the same origin o (or destination d) Decision variables: quantity of the super commodity k’ assigned to each “tree” or “sub-network” for k’ Formulations are equivalent 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Sample Network Arcs Commodities i j cost capy # o d quant 1 2 1 20 4 b 3 e Commodities # o d quant 1 1 3 5 2 1 4 15 3 2 4 5 4 3 4 10 Arcs i j cost capy 1 2 1 20 1 3 2 10 2 3 3 20 2 4 4 10 3 4 5 40 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Notation Parameters Decision Variables A: set of all network arcs K: set of all commodities N: set of all network nodes O(k) [D(k)]: origin [destination] node for commodity k cijk : per unit cost of commodity k on arc ij uij : total capacity on arc ij (assume uijk is unlimited for each k and each ij) dk : total quantity of commodity k Decision Variables xijk : number of units of commodity k assigned to arc ij 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Node-Arc Formulation 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Additional Notation Parameters Decision Variables Pk: set of all paths for commodity k, for all k cp : per unit cost of commodity k on path p = ij p cijk ijp : = 1 if path p contains arc ij; and = 0 otherwise Decision Variables fp: fraction of total quantity of commodity k assigned to path p 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

O/D Based Path Formulation å Î k P p f C d Minimize subject to d f u i j A k p ij P Î å £ " ( , ) : Bundle constraints f k K p P Î å = " 1 : Flow balance constraints f p P k K ³ " Î , : Non-neg. constraints 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Additional Notation Parameters Decision Variables S: set of source nodes nN for all commodities Qs: the set of all sub-networks originating at s TCqs: total cost of sub-network q originating at s pq : = 1 if path p is contained in sub-network q; and = 0 otherwise Decision Variables Rqs : fraction of total quantity of the super commodity originating at s assigned to sub-network q 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Sub-network Formulation q sÎ Q k P p R d C s å Î ) ( z Minimize s S subject to å Î " £ s ij Q q p k P A j i u R d ) , ( sz : Capacity Limits on Each Arc R s s S q Q Î å = " 1 : Mass Balance Requirements R s q Q s S ³ " Î , : Nonnegative Path Flow Variables 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Linear MCF Problem Solution Obvious Solution LP Solver Difficulty Problem Size: (|N|=|Nodes|, |C|=|Commodities|, |A|=|Arcs|) Node-arc formulation: Constraints: |N|*|C| + |A| Variables: |A|*|C| Path formulation: Constraints: |A| + |C| Variables: |Paths for ALL commodities| Sub-network formulation: Constraints: |A|+|Origins| Variables: |Combinations of Paths by Origin| 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

General MCF Solution Strategy Try to Decompose a Hard Problem Into a Set of Easy Problems 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

MCF Solution Procedures I Partitioning Methods Exploit Network Structure to Speed Up Simplex Matrix Computations Resource-Directive Decomposition Repeat until Optimal: Allocate Arc Capacity Among Commodities Find Optimal Flows Given Allocation 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

MCF Solution Procedures II Price-Directive Decomposition Repeat until Optimal: Modify Flow Cost on Arc Ignore Bundle Constraints, Find Optimal Flows 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Revisiting the Path Formulation MINIMIZE  k K  pPk dk cp fp subject to: pPk  k K dk fpijp  uij  ijA pP(k) fp = 1  kK fp  0  pPk,  kK 21-Nov-18 1.224J/ESD.204J

By-products of the Simplex Algorithm: Dual Variable Values Duals -ij: the dual variable associated with the bundle constraint for arc ij ( is non-negative) k : the dual variable associated with the commodity constraints Economic Interpretation  ij : the value of an additional unit of capacity on arc ij  k/dk : the minimal cost to send an additional unit of commodity k through the network 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J 1 1

Modified Costs Definition: Modified cost for arc ij and commodity k = cijk+ij Definition: Modified cost for path p and commodity k = ijA (cijk + ij )ijp 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J 1 1

Optimality Conditions for the Path Formulation f*p and *ij , *k are optimal for all k and all ij iff: Primal feasibility is satisfied pPk  k K dk f*pijp  uij  ijA pP(k) f*p = 1  kK f*p  0  pPk,  kK Complementary slackness is satisfied *ij(pPk  k K dk f*pijp - uij ) = 0,  ijA *k (p Pk f*p – 1) = 0,  kK Dual feasibility is satisfied (reduced cost is non-negative for a minimization problem) (dk cp +  ij A dk ij ijp ) - k = dk (  ij A (cijk + ij) ijp - k /dk )  0,  p Pk,  k K 21-Nov-18 1.224J/ESD.204J

Multi-commodity Flow Optimality Conditions The price for an additional unit of capacity is 0 unless capacity is fully utilized *ij(pPk  k K dk f*pijp - uij ) = 0,  ijA A path p for commodity k is utilized only if its “modified cost” (that is, ijA (cijk + *ijijp)) is minimal, for all paths pPk Reduced Costs all non-negative: c’p = dk (  ij A (cijk + *ij) ijp - *k /dk )  0,  p Pk,  k K f*p (ijA (cijk + *ij ) ijp - *k /dk ) = 0, 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J 3

Column Generation- A Price Directive Decomposition Millions/Billions of Variables Restricted Master Problem (RMP) Constraints Never Considered Start Added 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

RMP and Optimality Conditions Consider f*p and *ij , *k optimal for RMP, then Primal feasibility is satisfied pPk  k K dk f*pijp  uij  ijA pP(k) f*p = 1  kK f*p  0  pPk,  kK Complementary slackness is satisfied *ij(pPk  k K dk f*pijp - uij ) = 0,  ijA *k (p Pk f*p – 1) = 0,  kK Dual feasibility is guaranteed (reduced cost is non-negative) ONLY for a path p included in RMP (dk cp +  ij A dk ij ijp ) - k = dk (  ij A (cijk + ij) ijp - k /dk )  0,  p Pk,  k K 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J 3

LP Solution: Column Generation Step 1: Solve Restricted Master Problem (RMP) with subset of all variables (columns) Step 2: Solve Pricing Problem to determine if any variables when added to the RMP can improve the objective function value (that is, if any variables have negative reduced cost) Step 3: If variables are identified in Step 2, add them to the RMP and return to Step 1; otherwise STOP 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J 10

Pricing Problem Or, equivalently: min p Pk  ij A (cijk + ij) ijp Given  (non-negative) and k (unrestricted), the optimal duals for the current restricted master problem, the pricing problem, for each p Pk, k K is min p Pk (dk (  ij A (cijk + ij) ijp - k /dk ) Or, equivalently: min p Pk  ij A (cijk + ij) ijp A shortest path problem for commodity k (with modified arc costs) 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Example- Iteration 1 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Example- Iteration 2 NOTE: Next year, it would be good to supplement this with finding the paths with negative reduced costs by finding the shortest paths on the network, using modified costs. NOTE: It might be a good idea to also cover lagrangian relaxation and approximation algorithms for mcf problems 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

MCF Optimality Conditions For each pPk, for each k, the reduced cost cp: cp = (dk cp +  ij A dk ij ijp ) - k = ij (dkcijk + dkij)ijp - k = ij (cijk + ij)ijp - k /dk  0 where ,  are the optimal duals for the current restricted master problem cp = 0, for each utilized path p implies ij (dkcijk + dkij) ijp = k or equivalently, ij (cijk + ij) ijp = k/dk So if, minpP(k) cp = ij (cijk + ij) ijp* - k/dk  0, the current solution to the restricted master problem is optimal for the original problem If minpP(k) cp = ij (cijk + ij) ijp* - k/dk < 0, add p* to restricted master problem 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Data Set Data Set Constraint Matrix Size 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Computational Results Number of Nodes: 807 Number of Links: 1,363 Number of Commodities: 17,539 Computational Result (IBM RS6000, Model 370) Path Model: 44 minutes Sub-network Model: < 1 minute 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

LP Computational Experiment Test effect of adding most negative reduced cost column for each commodity vs. adding several negative reduced cost columns for each commodity 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Generating Several Columns Per Commodity Select any basic column (fp has reduced cost = 0) for some path p and commodity k, call it the key(k) Add all simple paths representing symmetric difference between most negative reduced cost path and key(k) 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Example p1- most negative reduced cost path for k key(k) Add to LP: 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

LP Solution: One Path per Commodity 301 nodes, 497 arcs, 1320 commodities. Times are on an IBM RS6000/590. 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

LP Solution: All Simple Paths for Each Commodity 301 nodes, 497 arcs, 1320 commodities. Times are on an IBM RS6000/590. 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Integer Multi-Commodity Network Flows Consider the modified multi-commodity network flow problem: Added integrality restriction that each commodity must be assigned to exactly one path fp (0.1),  p  Pk Solution procedure: branch-and-bound, specialized to handle large-scale problems 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Integer Multicommodity Flows: Problem Formulation MINIMIZE  k K  pPk dk cp fp subject to: pPk  k K dk fpijp  uij  ijA pP(k) fp = 1  kK fp  (0,1)  pPk,  kK 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Branch-and-Bound: A Solution Approach for Binary Integer Programs All possible solutions at leaf nodes of tree (2n solutions, where n is the number of variables) 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Branch-and-Bound: A Solution Approach for Binary Integer Programs Branch-and-Bound is a smart enumeration strategy: With branching, all possible solutions (e.g., 2number of path for all commodities) are enumerated With bounding, only a (usually) small subset of possible solutions are evaluated before a provably optimal solution is found 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Bounding: The Linear Programming (LP) Relaxation Consider the linear path-based MCF problem formulation Objective is to minimize The LP relaxation replaces fp  0,1 with 1  fp  0 Let zLP* represent the optimal LP solution and let zIP* represent the optimal IP solution zLP*  zIP* LP’s provide a bound on the lowest possible value of the optimal integer solution 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Branching Consider an IP with binary restrictions on all variables, denoted P(1) Let LP(1) denote the linear programming relaxation of P(1) and let x*(1) denote the optimal solution to LP(1) If there is no variable with fractional value in x*(1), x*(1) solves (is optimal for) P(1) If there is at least one variable with fractional value in x*(1), call it xl*(1), then any optimal solution for P(1) has xl*(1)=0 or xl*(1)=1 Left branch: xl*(1)=0 Right branch: xl*(1)=1 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

A Pictorial View feasible IP feasible LP 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Relationship between Bound and Tree Depth Let x*(1) be the optimal solution to LP(1) with at least one fractional variable xl*(1) Let the optimal solution value for LP(1) be denoted z*(1) Let LP(2) = LP(1) + [xl*(1) = 0 or xl*(1) = 1] Let the optimal solution value for LP(2) be denoted z*(2) Then z*(1)  z*(2) 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Tree Pruning Incumbent: Current best feasible (IP) solution value = zIP 1 x1 = 1 x1=0 2 3 x2=1 x2=0 x2=1 x2=0 4 5 6 7 x3=1 x3=0 x3=1 x3=0 If z*(LP(2)) zIP, PRUNE (FATHOM) tree at node 2 (solutions on the LHS of tree cannot be optimal. 1/2 of the solutions (nodes) do not need to be evaluated!) If z*(LP(2)) is integral, PRUNE tree at node 2 (solutions in sub-tree at node 2 cannot be better.) If LP(2)is infeasible, PRUNE tree at node 2 (solutions in sub-tree at node 2 cannot be feasible.) 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Branch-and-Bound Algorithm Beginning with rootnode (minimization): Bound: Solve the current LP with this and all restrictions along the (back) path to the rootnode enforced Prune: If optimal LP value is greater than or equal to the incumbent solution: Prune If LP is infeasible: Prune If LP is integral: Prune and update incumbent solution Branch: Set some variable to an integer value Repeat until all nodes pruned 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Branch-and-Price Solution Approach Branch-and-bound tailored to solve large-scale integer programs Bounding Solve LP using column generation at each node of the branch-and-bound tree Branching New columns might have to be generated to find an optimal solution to the constrained problem Want to design the branching decision so that the algorithm for the pricing is unchanged as the branch-and-bound tree is processed 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Example Revisited NOTE: Next year, it would be good to supplement this with finding the paths with negative reduced costs by finding the shortest paths on the network, using modified costs. NOTE: It might be a good idea to also cover lagrangian relaxation and approximation algorithms for mcf problems 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Branch-and-Price: Branching and Compatibility with the Pricing Problem Branching decision for commodity k, fp = 1: No pricing problem solution is necessary All other variables for k are removed from the model Branching decision for commodity k, fp = 0: The solution to the pricing problem (a shortest path problem) CANNOT generate path p as the shortest path, must instead find the next shortest path In general, at nodes of depth l in the branch-and-bound tree, the pricing problem must potentially generate the kth shortest path 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

An Alternative Branching Idea: Branch on Small Decisions Consider commodity k whose flow is split Assume k takes 2 paths, p1 and p2 Let d be the divergence node p1 o(k) d(k) p2 d 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Divergence Node Let a1 be the arc out of d on p1 and a2 be the arc out of d on p2 A(d) = {a1, a2, a3, a4}, A(d,a1) = {a1,a3}, A(d, a2) = {a2, a4} d a2 a4 a1 a3 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Branching Rule Create two branches, one where And the other with 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Branch-and-Bound Results: Conventional Branching Rule Eight telecommunications test problems 50 nodes, 130 arcs, 585 commodities Computational experiment on an IBM RS6000/590 For each of the eight test problems, run time of 3600 seconds No feasible solution was found 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Branch-and-Bound Results: Our New Branching Rule All test problems have 50 nodes, 130 arcs, 585 commodities. Run times on an IBM RS6000/590. 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Conclusions I Choose your formulation carefully Trade-off memory requirements and solution time Sub-network formulation can be effective when low level of congestion in the network Problem size often mandates use of combined column and row generation 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J

Conclusions II Solution time is affected dramatically by The complexity of the pricing problem Exploitation of problem structure, pre-processing, LP solver selection, etc. Branching strategy should preserve the structure of the pricing problem Branch on “small” decisions, not the variables in the column generation formulation 11/21/2018 Barnhart 1.206J/16.77J/ESD.215J